Locally Symmetric Families of Curves and Jacobians

The moduli space A g of principally polarized abelian varieties of dimension g is a locally symmetric variety. Denote the closure in A g of the locus of jacobians by J g . In this paper we make a preliminary investigation of locally symmetric subvarieties X of A g that are contained in J 9 and contain the moduli point of the jacobian of a smooth curve. Under certain hypotheses (X is “simple”, the corresponding family of abelian varieties can be lifted to a family of curves and a rank condition), we prove that such an X has to be a ball quotient. Our main tools are group cohomology and naive geometric considerations.